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Register a new userSchur Rings over Z times Z_3
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For the direct product $cZtimes cZ_3$ of infinite cyclic group $cZ$ and a cyclic group $cZ_3$ of order $3$, the schur rings over it are classified. In particular, all the schur rings are proved to be traditional.
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The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over R that are not generalized wreath products.
Let R be a finitely generated commutative ring with 1, let A be an indecomposable 2-spherical generalized Cartan matrix of size at least 2 and M=M(A) the largest absolute value of a non-diagonal entry of A. We prove that there exists an integer n=n(A) such that the Kac-Moody group G_A(R) has property (T) whenever R has no proper ideals of index less than n and all positive integers less than or equal to M are invertible in R.
For every pair of distinct primes $p$, $q$ we prove that $mathbb{Z}_p^3 times mathbb{Z}_q$ is a CI-group with respect to binary relational structures.
For a positive integer $g$, let $mathrm{Sp}_{2g}(R)$ denote the group of $2g times 2g$ symplectic matrices over a ring $R$. Assume $g ge 2$. For a prime number $ell$, we give a self-contained proof that any closed subgroup of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$ which surjects onto $mathrm{Sp}_{2g}(mathbb{Z}/ellmathbb{Z})$ must in fact equal all of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.
The commuting graph of a group G, denoted by Gamma(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Z_m be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Z_m) to contribute to the conjecture that there is a universal upper bound on diam(Gamma(G)) for any finite group G when Gamma(G) is connected. For any composite m, it is shown that Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) are connected and diam(Gamma(GL(n,Z_m))) = diam(Gamma(M(n,Z_m))) = 3. For m a prime, the instances of connectedness and absolute bounds on the diameters of Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) when they are connected are concluded from previous results.
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